Theorem splcl 12687

Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)

Assertion

Ref	Expression
splcl	|- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Proof of Theorem splcl

Dummy variables s b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3122	. . . 4 |- ( S e. Word A -> S e. _V )
2		otex 4712	. . . 4 |- <. F , T , R >. e. _V
3		id 22	. . . . . . . 8 |- ( s = S -> s = S )
4		fveq2 5864	. . . . . . . . . 10 |- ( b = <. F , T , R >. -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
5	4	fveq2d 5868	. . . . . . . . 9 |- ( b = <. F , T , R >. -> ( 1st ` ( 1st ` b ) ) = ( 1st ` ( 1st ` <. F , T , R >. ) ) )
6	5	opeq2d 4220	. . . . . . . 8 |- ( b = <. F , T , R >. -> <. 0 , ( 1st ` ( 1st ` b ) ) >. = <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. )
7	3, 6	oveqan12d 6301	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) = ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) )
8		simpr 461	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> b = <. F , T , R >. )
9	8	fveq2d 5868	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` b ) = ( 2nd ` <. F , T , R >. ) )
10	7, 9	oveq12d 6300	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) = ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) )
11		simpl 457	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> s = S )
12	8	fveq2d 5868	. . . . . . . . 9 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 1st ` b ) = ( 1st ` <. F , T , R >. ) )
13	12	fveq2d 5868	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( 2nd ` ( 1st ` b ) ) = ( 2nd ` ( 1st ` <. F , T , R >. ) ) )
14	11	fveq2d 5868	. . . . . . . 8 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( # ` s ) = ( # ` S ) )
15	13, 14	opeq12d 4221	. . . . . . 7 |- ( ( s = S /\ b = <. F , T , R >. ) -> <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. = <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. )
16	11, 15	oveq12d 6300	. . . . . 6 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) = ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) )
17	10, 16	oveq12d 6300	. . . . 5 |- ( ( s = S /\ b = <. F , T , R >. ) -> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) concat ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
18		df-splice 12509	. . . . 5 |- splice = ( s e. _V , b e. _V |-> ( ( ( s substr <. 0 , ( 1st ` ( 1st ` b ) ) >. ) concat ( 2nd ` b ) ) concat ( s substr <. ( 2nd ` ( 1st ` b ) ) , ( # ` s ) >. ) ) )
19		ovex 6307	. . . . 5 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. _V
20	17, 18, 19	ovmpt2a 6415	. . . 4 |- ( ( S e. _V /\ <. F , T , R >. e. _V ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
21	1, 2, 20	sylancl 662	. . 3 |- ( S e. Word A -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
22	21	adantr 465	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) )
23		swrdcl 12605	. . . . 5 |- ( S e. Word A -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
24	23	adantr 465	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A )
25		ot3rdg 6797	. . . . . 6 |- ( R e. Word A -> ( 2nd ` <. F , T , R >. ) = R )
26	25	adantl 466	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) = R )
27		simpr 461	. . . . 5 |- ( ( S e. Word A /\ R e. Word A ) -> R e. Word A )
28	26, 27	eqeltrd 2555	. . . 4 |- ( ( S e. Word A /\ R e. Word A ) -> ( 2nd ` <. F , T , R >. ) e. Word A )
29		ccatcl 12554	. . . 4 |- ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) e. Word A /\ ( 2nd ` <. F , T , R >. ) e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A )
30	24, 28, 29	syl2anc 661	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A )
31		swrdcl 12605	. . . 4 |- ( S e. Word A -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
32	31	adantr 465	. . 3 |- ( ( S e. Word A /\ R e. Word A ) -> ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A )
33		ccatcl 12554	. . 3 |- ( ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) e. Word A /\ ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
34	30, 32, 33	syl2anc 661	. 2 |- ( ( S e. Word A /\ R e. Word A ) -> ( ( ( S substr <. 0 , ( 1st ` ( 1st ` <. F , T , R >. ) ) >. ) concat ( 2nd ` <. F , T , R >. ) ) concat ( S substr <. ( 2nd ` ( 1st ` <. F , T , R >. ) ) , ( # ` S ) >. ) ) e. Word A )
35	22, 34	eqeltrd 2555	1 |- ( ( S e. Word A /\ R e. Word A ) -> ( S splice <. F , T , R >. ) e. Word A )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   <.cop 4033   <.cotp 4035   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   0cc0 9488   #chash 12369  Word cword 12496   concat cconcat 12498   substr csubstr 12500   splice csplice 12501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-substr 12508  df-splice 12509

This theorem is referenced by:  psgnunilem2  16316  efglem  16530  efgtf  16536  frgpuplem  16586